669 research outputs found
Asymptotically optimal importance sampling for Jackson networks with a tree topology
This note describes an importance sampling (IS) algorithm to estimate buffer overflows of stable Jackson networks with a tree topology. Three new measures of service capacity and traffic in Jackson networks are introduced and the algorithm is defined in their terms. These measures are effective service rate, effective utilization and effective service-to-arrival ratio of a node. They depend on the nonempty/empty states of the queues of the network. For a node with a nonempty queue, the effective service rate equals the node's nominal service rate. For a node i with an empty queue, it is either a weighted sum of the effective service rates of the nodes receiving traffic directly from node i, or the nominal service rate, whichever smaller. The effective utilization is the ratio of arrival rate to the effective service rate and the effective service-to-arrival ratio is its reciprocal. The rare overflow event of interest is the following: given that initially the network is empty, the system experiences a buffer overflow before returning to the empty state. Two types of buffer structures are considered: (1) a single system-wide buffer shared by all nodes, and (2) each node has its own fixed size buffer. The constructed IS algorithm is asymptotically optimal, i. e., the variance of the associated estimator decays exponentially in the buffer size at the maximum possible rate. This is proved using methods from (Dupuis et al. in Ann. Appl. Probab. 17(4): 1306-1346, 2007), which are based on a limit Hamilton-Jacobi-Bellman equation and its boundary conditions and their smooth subsolutions. Numerical examples involving networks with as many as eight nodes are provided
Efficient simulation of large deviation events for sums of random vectors using saddle-point representations
We consider the problem of efficient simulation estimation of the
density function at the tails, and the probability of large
deviations for a sum of independent, identically distributed (i.i.d.),
light-tailed and nonlattice random vectors. The latter problem
besides being of independent interest, also forms a building block
for more complex rare event problems that arise, for instance, in
queuing and financial credit risk modeling. It has been extensively
studied in the literature where state-independent, exponential-twisting-based
importance sampling has been shown to be asymptotically
efficient and a more nuanced state-dependent exponential twisting
has been shown to have a stronger bounded relative error property.
We exploit the saddle-point-based representations that exist for
these rare quantities, which rely on inverting the characteristic
functions of the underlying random vectors. These representations
reduce the rare event estimation problem to evaluating certain
integrals, which may via importance sampling be represented as
expectations. Furthermore, it is easy to identify and approximate the
zero-variance importance sampling distribution to estimate these
integrals. We identify such importance sampling measures and show
that they possess the asymptotically vanishing relative error
property that is stronger than the bounded relative error
property. To illustrate the broader applicability of the proposed
methodology, we extend it to develop an asymptotically vanishing
relative error estimator for the practically important expected
overshoot of sums of i.i.d. random variables
Efficient simulation of density and probability of large deviations of sum of random vectors using saddle point representations
We consider the problem of efficient simulation estimation of the density
function at the tails, and the probability of large deviations for a sum of
independent, identically distributed, light-tailed and non-lattice random
vectors. The latter problem besides being of independent interest, also forms a
building block for more complex rare event problems that arise, for instance,
in queueing and financial credit risk modelling. It has been extensively
studied in literature where state independent exponential twisting based
importance sampling has been shown to be asymptotically efficient and a more
nuanced state dependent exponential twisting has been shown to have a stronger
bounded relative error property. We exploit the saddle-point based
representations that exist for these rare quantities, which rely on inverting
the characteristic functions of the underlying random vectors. These
representations reduce the rare event estimation problem to evaluating certain
integrals, which may via importance sampling be represented as expectations.
Further, it is easy to identify and approximate the zero-variance importance
sampling distribution to estimate these integrals. We identify such importance
sampling measures and show that they possess the asymptotically vanishing
relative error property that is stronger than the bounded relative error
property. To illustrate the broader applicability of the proposed methodology,
we extend it to similarly efficiently estimate the practically important
expected overshoot of sums of iid random variables
Linear Stochastic Fluid Networks: Rare-Event Simulation and Markov Modulation
We consider a linear stochastic fluid network under Markov modulation, with a
focus on the probability that the joint storage level attains a value in a rare
set at a given point in time. The main objective is to develop efficient
importance sampling algorithms with provable performance guarantees. For linear
stochastic fluid networks without modulation, we prove that the number of runs
needed (so as to obtain an estimate with a given precision) increases
polynomially (whereas the probability under consideration decays essentially
exponentially); for networks operating in the slow modulation regime, our
algorithm is asymptotically efficient. Our techniques are in the tradition of
the rare-event simulation procedures that were developed for the sample-mean of
i.i.d. one-dimensional light-tailed random variables, and intensively use the
idea of exponential twisting. In passing, we also point out how to set up a
recursion to evaluate the (transient and stationary) moments of the joint
storage level in Markov-modulated linear stochastic fluid networks
Adaptive importance sampling technique for markov chains using stochastic approximation
For a discrete-time finite-state Markov chain, we develop an adaptive importance sampling scheme to estimate the expected total cost before hitting a set of terminal states. This scheme updates the change of measure at every transition using constant or decreasing step-size stochastic approximation. The updates are shown to concentrate asymptotically in a neighborhood of the desired zero-variance estimator. Through simulation experiments on simple Markovian queues, we observe that the proposed technique performs very well in estimating performance measures related to rare events associated with queue lengths exceeding prescribed thresholds. We include performance comparisons of the proposed algorithm with existing adaptive importance sampling algorithms on some examples. We also discuss the extension of the technique to estimate the infinite horizon expected discounted cost and the expected average cost
Variance Reduction Techniques in Monte Carlo Methods
Monte Carlo methods are simulation algorithms to estimate a numerical quantity in a statistical model of a real system. These algorithms are executed by computer programs. Variance reduction techniques (VRT) are needed, even though computer speed has been increasing dramatically, ever since the introduction of computers. This increased computer power has stimulated simulation analysts to develop ever more realistic models, so that the net result has not been faster execution of simulation experiments; e.g., some modern simulation models need hours or days for a single ’run’ (one replication of one scenario or combination of simulation input values). Moreover there are some simulation models that represent rare events which have extremely small probabilities of occurrence), so even modern computer would take ’for ever’ (centuries) to execute a single run - were it not that special VRT can reduce theses excessively long runtimes to practical magnitudes.common random numbers;antithetic random numbers;importance sampling;control variates;conditioning;stratied sampling;splitting;quasi Monte Carlo
Network Tomography: Identifiability and Fourier Domain Estimation
The statistical problem for network tomography is to infer the distribution
of , with mutually independent components, from a measurement model
, where is a given binary matrix representing the
routing topology of a network under consideration. The challenge is that the
dimension of is much larger than that of and thus the
problem is often called ill-posed. This paper studies some statistical aspects
of network tomography. We first address the identifiability issue and prove
that the distribution is identifiable up to a shift parameter
under mild conditions. We then use a mixture model of characteristic functions
to derive a fast algorithm for estimating the distribution of
based on the General method of Moments. Through extensive model simulation and
real Internet trace driven simulation, the proposed approach is shown to be
favorable comparing to previous methods using simple discretization for
inferring link delays in a heterogeneous network.Comment: 21 page
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